PHYSICAL REVIEW E 92, 013205 (2015)

Lattice mechanics of origami tessellations

Arthur A. Evans,1,* Jesse L. Silverberg,2 and Christian D. Santangelo1

1Department of Physics, UMass Amherst, Amherst, Massachusetts 01003, USA2Department of Physics, Cornell University, Ithaca, New York 14853, USA

(Received 18 March 2015; published 27 July 2015)

Origami-based design holds promise for developing materials whose mechanical properties are tuned by creasepatterns introduced to thin sheets. Although there have been heuristic developments in constructing patterns withdesirable qualities, the bridge between origami and physics has yet to be fully developed. To truly considerorigami structures as a class of materials, methods akin to solid mechanics need to be developed to understandtheir long-wavelength behavior. We introduce here a lattice theory for examining the mechanics of origamitessellations in terms of the topology of their crease pattern and the relationship between the folds at each vertex.This formulation provides a general method for associating mechanical properties with periodic folded structuresand allows for a concrete connection between more conventional materials and the mechanical metamaterialsconstructed using origami-based design.

DOI: 10.1103/PhysRevE.92.013205 PACS number(s): 46.25.Cc, 05.50.+q, 62.20.−x, 81.05.Zx

I. INTRODUCTION

While for hundreds of years origami has existed as anartistic endeavor, recent decades have seen the application offolding thin materials to the fields of architecture, engineering,and material science [1–7]. Controlled actuation of thin mate-rials via patterned folds has led to a variety of self-assemblystrategies in polymer gels [8] and shape-memory materials [4],as well elastocapillary self-assembly [9], leading to the designof a new category of shape-transformable materials inspiredby origami design. The origami repertoire itself, buoyed by ad-vances in the mathematics of folding and the burgeoning fieldof computational geometry [10], is no longer limited to designsof animals and children’s toys that dominate the art in popularconsciousness, but now includes tessellations, corrugations,and other nonrepresentational structures whose mechanicalproperties are of interest from a scientific perspective. Theseproperties originate from the confluence of geometry andmechanical constraints that are an intrinsic part of origami andultimately allow for the construction of mechanical metamate-rials using origami-based design [1–4,6,11–13]. In this paperwe formulate a general theory for periodic lattices of folds inthin materials and combine the language of traditional latticesolid mechanics with the geometric theory underlying origami.

A distinct characteristic of all thin materials is that geo-metric constraints dominate the mechanical response of thestructure. Because of this strong coupling between shape andmechanics, it is far more likely for a thin sheet to deform bybending without stretching. Strategically weakening a materialwith a crease or fold, and thus lowering the energetic cost ofstretching, allows complex deformations and reordering ofthe material for negligible elastic energy cost. This vanishingenergy cost, especially combined with increased control overmicroscopic and nanoscopic material systems, indicates greatpromise for structures whose characteristics depend primarilyon geometry, rather than material composition.

By patterning creases, hinges, or folds into an otherwiseflat sheet (be it composed of paper, metal, or polymer gel), the

*artio.evans@gmail.com

bulk material is imbued with an effective mechanical response.In contrast to conventional composites engineering, whereinmethods generally rely on designing response based on theinteraction between the constituent parts that compose thematerial, origami-based design injects novelty at the atomiclevel; even single vertices of origami behave as engineeringmechanisms [14], providing novel functionality such as com-plicated bistability [15–17] and auxetic behavior [6,11–13].This generic property inspires the identification of origamitessellations with mechanical metamaterials or a compositewhose effective properties arise from the structure of the unitcell. Although originally introduced to guide electromagneticwaves [18], rationally designed mechanical metamaterialshave since been developed that control wave propagationin acoustic media [19,20], thin elastic sheets, and curvedshells [21–24] and harness elastic instabilities to generateauxetic behavior [25–29].

Traditional metamaterials invoke the theory of linearresponse in wave systems, but currently there is no generaltheory for predicting the properties of origami-inspired designson the basis of symmetry and structure. In the following wepropose a general framework for analyzing the kinematics andmechanics of an origami tessellation as a crystalline material.By treating a periodic crease pattern, we naturally connect thegeometric mathematics of origami to the more conventionalanalysis of elasticity in solid state lattice structures. In Sec. IIwe outline the general formalism required to find the kinematicsolutions for a single origami vertex. In Sec. III we discuss thegeneral formulation for a periodic lattice, including both thekinematics of deformation modes and energetics for a periodiccrease pattern. In Sec. IV we examine the well-known casestudy of the Miura-ori pattern. Our analysis here recoversknown aspects of the Miura-ori pattern and identifies keyfeatures that have not been quantitatively discussed previously.

II. SINGLE ORIGAMI VERTEX

Many of the design strategies for self-folding materialsinvolves a single fold, an array of nonintersecting folds, oran array of folds that intersect only at the boundary of thematerial [9,30–33]. From a formal standpoint, we define a

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EVANS, SILVERBERG, AND SANTANGELO PHYSICAL REVIEW E 92, 013205 (2015)

FIG. 1. (Color online) (a) Graph for a single vertex. This degree-6 vertex has its graph determined by the six sector angles αi . Each creasehas a dihedral angle fi associated with it. In the flat case every fi = π or, equivalently, every fold angle is identically zero, since the fold angleis defined as the supplement of the dihedral angle. (b) By assigning fold angles to each crease, a three-dimensional embedding of the vertex (i.e.,the folded form of the origami) is fully determined. Every face must rotate rigidly about the defined creases and the sector angles must remainconstant. There is a limited set of fold angles that will solve these conditions. (c) Schematic projection of the curve of intersection betweenthe unit sphere and the folded form origami. For an N -degree vertex this projection generates a spherical N -gon. To proceed, the N -gon isdivided into N − 2 spherical triangles and the interior angles (i.e., the fi) follow as a result of applying the rules of spherical trigonometry. Allthree-dimensional origami structures are visualized using Tessellatica, a freely available online package for Mathematica [34].

fold as a straight line demarcating the boundary between twoflat sheets of unbendable, unstretchable material. These sheets,in isolation, are allowed to rotate around the fold so that thestructure behaves mechanically like a simple hinge. If the foldis produced by plastically deforming a piece of material, ratherthan functioning as a hinge the fold has a preferred angle andis more precisely called a crease. Herein we shall use the termsinterchangeably, since the kinematic motions of a fold and theenergetics involved for a crease can be described separately. Animportant, and arguably defining, characteristic of an origamistructure is that it requires that more than one fold meet ata vertex. While each fold individually allows for unrestrictedrigid body rotation of a sheet, geometrical constraints arisewhen several folds coincide at a vertex. These constraints arewhat provide origami structures with their mechanical noveltyand ultimately are why deployable structures and mechanicalmetamaterials display exotic and tunable properties.

A vertex of degree N is defined as a point where N

straight creases meet. Figure 1(a) shows the crease pattern fora schematic six-degree vertex, with sectors defined by planarangles αi . The three-dimensional folded form of this vertex isfound by supplying fold angles to each of the creases, subject tothe constraints mentioned previously [35,36]. This procedureis an exercise in spherical trigonometry.

One way to visualize the constraints is to surround eachvertex with a sphere and consider the intersection between itand the surface [Fig. 1(b)]. In this construction, the side lengthsof the spherical polygon are the angles between adjacent folds,which must remain fixed, and the dihedral fold angles are theinternal angles of the polygon on the sphere. Since an N -sidedpolygon has N − 3 continuous degrees of freedom, each vertexdoes as well. These N − 3 degrees of freedom can be thoughtof, for example, as the angles between a fixed fold and theremaining nonadjacent folds.

Starting with a general vertex containing dihedral anglesfi , we use spherical trigonometry to calculate these angles interms of the N − 3 degrees of freedom. To calculate f1 we

partition the angle into sectors by subdividing the sphericalN -gon into N − 2 triangles [Fig. 1(c)]. We label the anglesthat lead from f1 to fi as �i , where �1 = α1 and �N−1 = αN aresector angles. All the angles αi are spherical polygon edges andsince origami structures allow only isometric deformations,these angles are constant. The �i are the angles subtendedby drawing a geodesic on the encapsulating sphere from f1

to fi+1; expressions for relating the �i to the fold angles fi

are found by using the spherical law of cosines around thevertex [35]:

f1 =N−2∑i=1

cos−1

[cos αi+1 − cos �i+1 cos �i

sin �i+1 sin �i

], (1)

f2 = cos−1

[cos �2 − cos α1 cos α2

sin α1 sin α2

], (2)

fN = cos−1

[cos �N−2 − cos αN−1 cos αN

sin αN−1 sin αN

], (3)

fi = cos−1

[cos �i−2 − cos αi−1 cos �i−1

sin �i−1 sin αi−1

]

+ cos−1

[cos �i − cos αi cos �i−1

sin �i−1 sin αi

]. (4)

These expressions are essentially all that is required to deter-mine the folding of a single vertex, although the associatedsolutions are generically multivalued. These results imply thatthere are multiple branches of configuration space for anygiven spherical polygon.

To specify the internal state of each vertex we define anN − 3 component vector s. Given the internal state of a vertex,all N of the dihedral fold angles are determined, which wecollect in the vector f(s). In practice, computations are vastlysimplified by choosing the appropriate degrees of freedom;for example, for a degree-6 vertex of the type displayed inFig. 1, we choose s = {�3,f2,f6} and the fold vector is givenby f = {f1,f2,f3,f4,f5,f6}.

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III. GENERAL LATTICE THEORY

To determine the mechanical properties of an origamitessellation we begin by examining how many vertices areconnected together in a crease pattern. When constructing areal piece of origami, artists and designers specify “mountain”and “valley” creases in the pattern to encode instructions forhow the structure will fold. In our formulation we will treat thecrease pattern as a simple connected graph, where each uniquecrease is an edge that connects two vertices to one another.

A. Kinematically allowed deformations

In addition to the origami constraints discussed above fora single vertex, joining multiple vertices together generatesfurther constraints on the folds. Consider a crease pattern thatconsists of P vertices. Each vertex vp, with p ∈ {1, . . . ,P },has Np folds, collected in the vector fp = (f p

1 fp

2 · · · f p

Np)T . If

we collect all the folds into the vector F , given by

F =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

f 11

f 12...

f 1N1

f 21...

f 2N2

...f P

1...

f PNP

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (5)

then we have the following constraint equation for the folds:

DF = 0, (6)

where D is a sparse rectangular matrix that enforces thecondition that if two vertices vq,vp are adjacent and twofolds Fi ,Fj connect vq,vp, then Fi = Fj (see Fig. 2 foran example). This constraint enforces the connectivity of thegraph, since each unique crease clearly must have a compatiblefold angle associated with the vertices that connect it. Each

f11

f12

f13

f14

f24

f23

f22

f21

f11

f12

f13

f14 f2

4

f23

f22

f21

(a) (b)

FIG. 2. (Color online) (a) Two degree-4 vertices with labeledfolds. (b) Graph for the crease pattern consisting of these two verticescontains a single crease that is shared by both vertices. In this case theconstraint equation DF = 0 simply becomes the scalar relationshipf 1

1 = f 23 .

row of D corresponds to a fold connecting a pair of verticesin the origami tessellation while each column correspondsto a component of F . Analysis of this construction is theessence of origami mechanics and lies at the heart of thedifficulty in determining general properties of tessellations andcorrugations. Finding the null vectors of D amounts to findingall of the possible solutions for the fold angles and thus all ofthe kinematically allowed motions of the rigid origami. Whilecomputational methods have been developed for simulatingthe kinematics of origami and linkage structures [2,6,11,12],there has been no general analytical study that seeks to identifymechanical properties based solely on the crease pattern.

The functions F (s) are, in general, nonlinear. To proceedanalytically, we expand s about a state s0 that solves theconstraint equations. That is, if F (s0) = F0 then DF0 ≡ 0.A trivial choice for s0 has every entry identically equal toπ , indicating that the piece of origami is unfolded. The morecommon, and more interesting, scenario involves a folded statewhere the values of the internal vector s0 are known. Assumingthat such a state exists, we write s = s0 + δs, with δs a smallperturbation, and then have

DJδs ≡ Rδs = 0, (7)

where the Jacobian of the fold angles for each vertex J ≡∂F/∂s|s0 is a block diagonal matrix defining the smalldeviations from the ground state s0 and R is a rigidity matrixthat informs on the infinitesimal isometric deformations ofthe origami structure [37,38]. This formulation is convenientsince it separates the effects of the crease pattern topology(contained entirely in D) from the constrained motion of asingle vertex (contained entirely in J). We can thus solve foreach of these matrices individually.

To find D, we first exploit the periodicity of the lattice todecompose the vector F and matrix D in a Fourier basis suchthat F = ∑

n,m eiq·xFq + c.c. Here q is a two-dimensionalwave vector and x = na1 + ma2 is the two-dimensionalposition vector of the fundamental unit cell on the creasepattern lattice, where (n,m) indexes this position in terms ofthe lattice vectors a1,2. Since F ≈ Jδs and J is independentof the lattice position, we also have δs = ∑

n,m eiq·xδsq + c.c.,where Fq = Jδsq . In this representation the constraints givenin Eq. (6) are

D(q)Fq = D(q)Jδsq = 0. (8)

Now, instead of a matrix operation over all the vertices,the size of D(q) is vastly simplified. For a pattern with p

distinct vertices per unit cell, each of degree Np, D(q) is a∑p

i=1 (Ni/2) × ∑p

i=1 Ni matrix. In Fourier space, D(q) is thecomplex-valued constraint matrix for the graph of the unitcell vertices and folds. Specifically, each fold of the unit cellis represented by a row in D(q) having only two nonzeroentries. Those entries all have the form ±eiq·a1 , ± eiq·a2 ,±1,depending on whether the fold connects to an adjacent unitcell along a1,2 or is internal to the unit cell.

The formulation in terms of the matrix R(q) is completelygeneral for any origami tessellation. The rectangular matrixD(q) carries all of the topological information regarding thefold network, while the Jacobian J carries the informationabout the type of vertex that has been specified; J will be

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block diagonal with one block for each vertex of a unit cell,but does not depend on q for a regular tessellation.

B. Origami energetics

While the R matrix determines the kinematically isometricdeformation to leading order, these constraints are generallynot the end of the story for real materials. Creases in foldedpaper, thermoresponsive gels with programmed folding angles,and elastocapillary hinges all balance energetic considerationswith geometric constraints. In many cases these creases andhinges act as torsional springs, while the bending of faces haveadditional elastic energy content [7,39,40].

The energy associated with the entire structure may bewritten, to quadratic order in the dihedral vectors, as

E = 12 (F − F0)TA(F − F0), (9)

where A is a general stiffness matrix and F0 is a referencefold angle. For linear response this is the most generic formfor the energy. In the simplest of cases A is constant over thelattice and diagonal with respect to F ; this models each creaseas a torsional spring with uniform spring constant [7,13,40]. Asmall amplitude response is found by examining the origamistructure near the ground state, that is, when F = F0. Whenthe energy is expanded about the ground state E0 we find

E = E0 + 12δsT JTAJδs, (10)

or in the Fourier decomposition

E = LW

2

∑q

δs†qMδsq, (11)

where L is the length of the tessellation in the a1 direction,W is the width in the a2 direction, and M = JTAJ is amatrix operator that is independent of wave number. Since thenull space of R(q) will determine the modes of deformation,the solution to this problem lies in finding the kinematicallyallowed deformations and then any energetic description willsimply involve a change of basis to a system of deformationsthat diagonalize the operator M.

IV. MIURA-ORI

As an example of this formulation, we consider inhomoge-neous deformations of a particular origami metamaterial, theMiura-ori. First introduced as a framework for a deployablesurface, the design appears often in nature, from plantleaves [41] to animal viscera [42]. Additionally, theoreticalcalculations and experiments have suggested the Miura-ori asa canonical, origami-based, auxetic metamaterial [6,7,11–13].Its ubiquity may be related to its simplicity: The Miura-ori isdetermined from a single crease angle α and the mountain andvalley assignments of the pattern shown in Fig. 3. Conventionalorigami mathematics considers that each Miura-ori vertexis degree 4 and thus there is only one degree of freedom.However, casual experimentation with a real Miura-ori quicklydemonstrates that it has far more than one degree of freedom,indicating an array of soft modes enabled by the bending ofthe individual faces. This breakdown of the assumptions ofmathematical origami is well known and there are many creasepatterns that are mathematically impossible to fold that can in

θ+θ−

φ+

φ−

β+

β−

θ+θ−

φ+

φ−

β+

β−

(b)(a)

α

FIG. 3. (Color online) (a) While the crease pattern of a Miura-origenerally introduces only four folds per vertex, the bending of facesacts to allow two extra folds per vertex, so the crease pattern weconsider is a triangulated lattice. At each vertex the dihedral anglescontained in f are determined by specifying the state vector s andsatisfying the geometric constraints. (b) Single-vertex origami withenclosing sphere to visualize the constraints between f and s.

fact be done with little effort [43]. To incorporate these extradegrees of freedom into Miura-ori, we assume that there aretwo extra folds per vertex to account for face bending. While inthe extreme case of the creases being perfectly rigid these extrafolds would actually take the form of stretching ridges [44],many real applications involve fabrication processes that willallow the face to be well approximated as perfect bending.Each unit cell in the tessellation has four six-valent vertices(Fig. 3) so there are 12 degrees of freedom per unit cell. In thisexample the fold vector for the ith vertex is given by fi =(θ i

+,φi+,βi

+,θ i−,βi

−,φi−)T and the vector F = (f1 f2 f3 f4)T .

There are three degrees of freedom per vertex that define theinternal state s, which we parametrize using three angles: ε,the angle between folds labeled θ± in Fig. 3, and the anglesφ± representing the bending of the faces. Using the geometricrelationships between the angles [35], we find the generalnonlinear relationship for a single vertex and then expand aboutthe ground state s0 = {ε + δε,π + δφ+,π + δφ−} to find thematrix J; here ε ∈ [π − 2α,π + 2α]. This expansion naturallyfollows from assuming that the faces are nearly flat and that theMiura-ori has been folded into the standard configuration. TheJacobian J = diag(J0 −J0 J0 −J0) is a 24 × 12 diagonalblock matrix formed from four identical blocks

J0 =

⎛⎜⎜⎜⎜⎜⎝

A C C

0 1 0B C 0

−A 0 0B 0 C

0 0 1

⎞⎟⎟⎟⎟⎟⎠, (12)

where

A = cos α csc(ε/2)/√

sin2(ε/2) − cos2 α, (13)

B = sin(ε/2)/√

sin2(ε/2) − cos2 α, (14)

C = csc(α/2)/2. (15)

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LATTICE MECHANICS OF ORIGAMI TESSELLATIONS PHYSICAL REVIEW E 92, 013205 (2015)

x

y

123

45

67

89

1011

12

161718

1314 15

222324

1920 21

a

b

(a) (b)

(c) ei(qxa/2+qyb/2)

8

14

3

151322

ei(qxa/2) ei(qyb/2)

FIG. 4. (Color online) (a) Miura-ori, without the assignment of mountain and valley folds, has a simple directed graph structure with aunit cell composed of four vertices. By tessellating these four vertices, the entire pattern emerges. Note that the tessellation is rectangular,with lattice vectors a1 = ax and a2 = by. (b) Each vertex has six folds, labeled in the fashion shown here. (c) In Fourier space, translationsassociated with connecting these folds together throughout the tessellation merely amounts to a phase factor associated with the appropriatewave number and lattice vector. Shown on the left is translating in the x direction. The middle shows translating in the y direction. The rightshows that connecting the extra folds involves a diagonal translation across the unit cell. Note that the five internal folds have a phase factoridentically equal to one.

To calculate the constraint matrix, we note that there are 12unique folds per unit cell, so D(q) is a 12 × 24 rectangularmatrix. It has a row for each bond in Fig. 4 with twononzero columns indicating which folds of each vertex

are interconnected. For internal folds the constraint matrix hasa value of ±1, while folds that leave the unit cell have a phasefactor associated with it. For convenience of computation wehave symmetrized these phase factors, and the full matrix isgiven by

DT (q) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 0 0 0 0 0 0 0 00 eiqy/2 0 0 0 0 0 0 0 0 0 00 0 eiqy/2 0 0 0 0 0 0 0 0 00 0 0 −e−iqx/2 0 0 0 0 0 0 0 00 0 0 0 −1 0 0 0 0 0 0 00 0 0 0 0 −1 0 0 0 0 0 00 0 0 eiqx/2 0 0 0 0 0 0 0 00 0 0 0 0 0 eiqx/2+iqy/2 0 0 0 0 00 0 0 0 0 0 0 eiqy/2 0 0 0 0

−1 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 −1 0 0 00 0 0 0 0 0 0 0 0 eiq/2 0 00 0 0 0 0 0 0 0 0 0 −e−iq/2 00 0 0 0 0 0 −e−iqx/2−iqy/2 0 0 0 0 00 0 −e−iqy/2 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 10 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 −e−iqx/2 0 00 0 0 0 0 0 0 0 0 0 0 −10 −e−iqy/2 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 −e−iqy/2 0 0 0 00 0 0 0 0 0 0 0 0 0 eiqx/2 00 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 1 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

(16)

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EVANS, SILVERBERG, AND SANTANGELO PHYSICAL REVIEW E 92, 013205 (2015)

A. Bulk deformation

The combination D(q)J is square such that Eq. (7) has anontrivial solution whenever det[D(q)J] = 0. We nondimen-sionalize the wave number by the physical lengths of the latticevectors such that qxa → qx and qyb → qy and the resultingdispersion relation is

cos2 α

sin4(ε0/2)sin2(qx/2) + sin2(qy/2) = 0. (17)

The only real solution to this equation is q = 0, indicatingthat an infinite origami tessellation does not admit spatiallyinhomogeneous solutions; only uniform deformations areallowed. The null space of R is three dimensional here,corresponding to three uniform deformation modes of theMiura-ori. These zero modes are given by the vectors �i :

�I =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

100100100100

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, �II =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0−110

−110

−110

−11

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, �III =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−2CA

11011

−2CA

11011

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (18)

These infinitesimal deformations of the unit cell correspondto a uniform contraction, a twisting mode, and a saddlelikedeformation, respectively (see Fig. 5). To describe the kine-matics of deformation, all we require are the null vectors ofthe constraint equations, but for examining energy associatedwith the creases we need to calculate the eigenvalues ofthe matrix M = JT AJ. In general, it is not unreasonableto assume that a creased and folded Miura-ori will have acrease stiffness k that is approximately equal for all patterned

I

II

III

λ

Γ

I

II

III

(b)(a)

0.01 0.1 1 10 1000.001

0.01

0.1

1

10

100

1000

FIG. 5. (Color online) Shapes and energy eigenvalues for thethree uniform modes for ε = π/2 and α = π/3. (a) The three uniformnull vectors correspond to a uniform mode (I), a twisting mode (II),and a saddle mode (III). These are identical to the modes determinednumerically in previous studies [6,12]. (b) Eigenvalues associatedwith each of the three bulk modes as a function of face stiffness �.Note that over a wide range the softest mode is the twisting mode(II), since it involves purely face bending.

creases, but the energy scale for bending of the faces willdepend on the material properties of the structure [7]. Theenergy for bending can be treated as an effective torsionalspring constant kb and thus the energy can be written in termsof the ratio kb/k ≡ �. Nondimensionalizing the energy bykLxLy , we find the energy eigenvalues λ in terms of thenull vectors. Decomposing the internal variable deformationδs = ∑

i aiψi , where ψi = �i/|�i | is the normalized nullvector with i ∈ {I,II,III}, we write Eq. (11) as

E = LxLy

2aT Ma, (19)

a =⎛⎝ aI

aII

aIII

⎞⎠, (20)

M =

⎛⎜⎝

ψTI MψI ψT

I MψII ψTI MψIII

ψTII MψI ψT

II MψII ψTII MψIII

ψTIIIMψI ψT

IIIMψII ψTIIIMψIII

⎞⎟⎠. (21)

Each matrix element of M represents overlaps between thenull vectors ψi and the energy matrix M; only in exceptionalcircumstances will M be diagonal in the null basis. In generalit is given by

M =

⎛⎜⎜⎜⎝

2(A2 + B2) 0√

2(A−B)BC√C2+A2

0 C2 + � 0√2(A−B)BC√

C2+A2 0 �A2+(3A2−2BA+2B2)C2

A2+C2

⎞⎟⎟⎟⎠.

(22)

An example for when M is diagonal is given by α = π/3,ε =π/2 (see Fig. 5), for which M becomes

M =⎛⎝8 0 0

0 1 + � 00 0 2

3 (3 + �)

⎞⎠. (23)

Note that for this particular combination of parameters theuniform expansion mode has a flat stiffness over all ranges of� since there is no face bending for this deformation. In theregime where face bending is relatively inexpensive (� � 1),the out-of-plane deformation modes are correspondingly softerthan the uniform deformation. These results are in agreementwith previous numerical research done on the structuralmechanics of Miura-ori [6,11,12]. Should other values of (α,ε)be chosen, the energy matrix is not necessarily diagonal andthus eigensolutions mix the null vectors.

B. Inhomogeneous deformation

For a finite tessellation, the deformation is fundamen-tally different, since some folds reach the boundary and,consequently, do not yield constraints. Since the tessellationmechanics are determined by the allowable deformations,which are determined by the constraint equations, the presenceof free boundaries allows much more flexibility and theMiura-ori develops additional degrees of freedom. Theselocalized edge states are reminiscent of evanescent wavesin electromagnetism, boundary layers in elastic lattices [45],and Rayleigh surface waves [46]. Letting qx ≡ q (where q is

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LATTICE MECHANICS OF ORIGAMI TESSELLATIONS PHYSICAL REVIEW E 92, 013205 (2015)

FIG. 6. (Color online) Experimental observations of deformation localization in an 8 × 8 Miura-ori tessellation. (a) An undeformedMiura-ori shows a regular periodic pattern. Under (b) small deformations, (c) large deformations, and (d) in the presence of a pop-throughdefect (PTD) [7], the lattice distorts to accommodate the induced strain. (e) Qualitatively, the amount of deformation localization can be easilyseen by a simple image subtraction between the deformed and undeformed state. (f) Measuring strain along the horizontal axis as a function ofunit cell position n relative to the location of the disturbance shows a rapid decay for all three scenarios (points). For small and large amplitudes,the decays can be readily fit to an exponential function with decay length � [upper (red) and lower (black) lines], whereas for a PTD, the decaylength can be estimated to within 100%. Because the PTD induces an extensional distortion rather than a compression, the strain is oppositelysigned. The inset is a plot of the decay length against an approximate measure of the distortion wave vector q showing that the larger wavevector decays much more rapidly than the shorter wave vectors. Within error bars, this measurement is consistent with an inverse relationshipbetween decay length and wave vector. The solid line is the theoretical prediction from Eq. (24) for ε = π/2 and α = π/3.

real), Eq. (17) yields qy = ±iκ(q), where deformations decayaway from the boundaries of constant y with a length scale� ≡ 1/κ(q), with

�(q) = 1

2|sinh−1[cos α sin(q/2)/(sin2 ε/2)]| . (24)

This localization length is readily observed in deformationexperiments on Miura-ori sheets (see Fig. 6). Using laser-cutsheets of paper, an 8 × 8 Miura-ori is constructed by foldingthe whole sheet using a planar angle of α = π/3 into theground state given by ε = π/2 [Fig. 6(a)]. Inhomogeneousdeformations are created using both an external indenter toapply a displacement [Figs. 6(b) and 6(c)] and by placingreversible pop-through defects [Fig. 6(d)] [7]. The strain γn

at each unit cell n is measured such that γn = �wn/w, where�ww is the change in width of the nth cell and w is the averagewidth for an undisturbed cell. As shown in Fig. 6, the straindecays exponentially away from the indenter with a decaylength that is consistent (within error) with our theoreticalpredictions.

To examine these deformation modes more quantitatively,we return to the dispersion relation given by Eq. (17). There aretwo possible solutions to Eq. (17), corresponding to differentdecay directions, and thus the null space of R correspondingto each of these branches is two dimensional. We decompose

δs(x) into a sum of upward (in y) decaying and downwarddecaying modes

δs = eiqx[(u1χ1e−k(q)y + u2χ2e

−k(q)y)

+ (d1η1ek(q)y + d2η2e

k(q)y)] + c.c. (25)

The vectors χ1,2 correspond to the upward decaying modesand η1,2 the downward decaying modes. Note that, since thevalues of the angles must be real, η1,2(q) = χ1,2(−q). In thelong-wavelength limit, i.e., q � 1, we have

χ1 =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−CAq+CB(q−2i)AB

02q

−CAq+CB(q+2i)AB

00

CAq−CB(q−2i)AB

00

CAq+CB(q+2i)AB

02q

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, χ2 =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

− 2CA

00

− 2CA

−2iq

0

− 2CA

−2iq

0

− 2CA

00

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (26)

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EVANS, SILVERBERG, AND SANTANGELO PHYSICAL REVIEW E 92, 013205 (2015)

q

λ

λ

λ

I

II

III, IV

q

0.5 1.0 1.5 2.0 2.5 3.0

5

10

15

0.5 1.0 1.5 2.0 2.5 3.0

5

10

15

0.5 1.0 1.5 2.0 2.5 3.0

5

10

15

Γ = 0.1

Γ = 1

Γ = 10

Γ = 0.1

Γ = 1

Γ = 10

I

II

III,IV

(b)(a)

0.5 1.0 1.5 2.0 2.5 3.0

1

2

3

4

0.5 1.0 1.5 2.0 2.5 3.0

1

2

3

4

0.5 1.0 1.5 2.0 2.5 3.0

1

2

3

4

5

FIG. 7. (Color online) Eigenvalues and mode shapes as a function of wave number for a given �. (a) Shown on the left is the mode structurefor � = 0.1, 1, and 10, with ε = π/2 and α = π/3. At long wavelengths the saddle mode I is the stiffest for a wide range of q, since it involvesboth bending of the faces and deformation of the angles away from the reference state. Shown on the right is the mode structure for � = 0.1, 1,and 10, with ε = π/2 and α = 9π/20. The labeled modes are in ascending order from largest eigenvalue (mode I) to lowest eigenvalue (modeIV). (b) Visualization of the basic modes for q = π/6.

The null space, and thus the number of elementary excitations,for a finite-size Miura-ori is actually different than for thelimit q → 0. While this may seem counterintuitive, the natureof the null vectors is inherently chiral, as indicated by thedecomposition into upward and downward decaying solutions.At q = 0, the dimensionality of the null space is smallerbecause there is no distinction between handedness for uniformdeformation.

C. Miura-ori’s soft modes

The vectors χ1,2 govern the kinematic deformations ofMiura-ori, giving the possible solutions to the constraintequations. For a tessellation with an associated torsional springenergy at each crease, the energy density per mode may bewritten in Fourier space as

E = LxLy

2c†(q)H(q)c(q), (27)

where

c(q) =

⎛⎜⎜⎜⎝

u1(q)

u2(q)

d1(q)

d2(q)

⎞⎟⎟⎟⎠ (28)

and H is the 2 × 2 Hermitian block matrix

H =(

H0 H1

H†1 H†

0

). (29)

The two independent blocks of H are given by

H0 =(

χ†1Mχ1 χ

†1Mχ2

χ†2Mχ1 χ

†2Mχ2

)(30)

and

H1 =(

χ†1Mη1 χ

†1Mη2

χ†2Mη1 χ

†2Mη2

). (31)

For finite wave number there are four modes of deformation.Typical eigenvalues of H(q) are shown in Fig. 7. The largesttwo eigenvalues are typically associated with changing ε,since there is an energetic cost even for very small �. Thetypically smallest two eigenvalues correspond to twistingmode and a fourth mode that has no analog in the zerowave number case. This mode has a qualitative shape thatis similar to the twisting mode and an energy that vanishesas q → 0, much like an acoustic mode in a crystal. Previousanalyses of inhomogeneous deformations have not found thismode, which we identify here as arising from the breakingof continuous symmetry when a boundary is added to oneside of the tessellation. The acoustic mode corresponds toan antisymmetric combination of upward and downwarddecaying modes; consequently, as q becomes smaller, thechange in fold angles associated with the combination cancelsand only three modes appear at q = 0.

The modes that are softest depend not only on thestiffness of face bending, but on the ground state definedby ε0 (see Fig. 7). This stiffness dependence is in accord

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LATTICE MECHANICS OF ORIGAMI TESSELLATIONS PHYSICAL REVIEW E 92, 013205 (2015)

with the previously predicted anisotropic in-plane stiffnessresponse [6,13]. Additionally, since our analysis allows forarbitrary size and wave number, we are able to capture theresponse of the previously unidentified acoustic mode.

V. DISCUSSION

While there has been numerical analysis of tessellationsin the past, our theoretical formulation provides severalkey insights into the design and understanding of origamimechanics. We not only analytically calculated expressionsfor first-order inhomogeneous deformations, but we foundan additional acoustic mode of deformation that has notbeen identified using numerics. Moreover, we have found ananalytical expression for a decay length that arises in Miura-oriand identified that these soft modes are edge states that cannotoccur in an infinite tessellation. Indeed, the appearance ofa single decay length and the ability to fully quantify thedeformation modes using a single wave number indicates thatthe boundaries of Miura-ori fully define the deformation state.We can directly conclude from this that, unlike normal solids,the number of degrees of freedom scales with the perimeter ofa finite tessellation, rather than the area. This result suggeststhat there are surface boundary states that can be used toprobe the full deformation of the material and hints at theconnection between our work and recent studies on topologicalmechanics [38]. In fact, our mathematical formalism sharesmany parallels with the topological mechanics of linkages[47–49], as well as the more conventional literature concerningtopological insulators and semimetals [50–52]. It remains tobe seen exactly how the symmetry and topology of the creasepattern affect the nature of chiral modes in origami, but thereis evidence to suggest that even slight modifications of the

crease pattern symmetry may lead to preferentially directedchiral states.

A great deal of this analysis can be carried through toother origami fold patterns. What is less clear, however, ishow the number of degrees of freedom, the null space ofR(q), changes for different fold patterns. At the outset it mayseem coincidental that the matrix R(q) is square. In fact, thisbehavior is likely more generic. In particular, the Miura-ori,with additional folds across the faces, is composed of triangularsubunits. In any triangulated origami fold pattern, vertices willtend to have, on average, six folds. Hence, for V vertices (withV very large), we have 3V unique folds and 3V degrees offreedom per vertex. Consequently, R(q) will be a 3V × 3V

square matrix for sufficiently large V .Finally, a great advantage to this approach is the ability

to separate the topological nature of the crease pattern fromthe geometry of the vertex. The ability to isolate mechanicaldeformations or elementary excitations in exotic materials isof great interest in quantum condensed matter [38], amorphoussolids [53–55], and complex fluids [56]. Our theoreticalframework for origami tessellations bridges the gap betweenthe origami mechanics literature and a theory of origamimetamaterials by identifying the constraint-based nature ofthe folding mechanisms and applying well-known methods ofanalysis from solid state physics and lattice mechanics.

ACKNOWLEDGMENTS

The authors acknowledge interesting and helpful discus-sions with Tom Hull, Robert Lang, Tomohiro Tachi, ScottWaitukaitis, Martin van Hecke, and Michael Assis. We alsothank F. Parish for help with the laser cutter. This work wasfunded by the National Science Foundation through Grant No.EFRI ODISSEI-1240441.

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